CLASSIFYING STARLIKE BODIES
Daniel Azagra and Tadeusz Dobrowolski
Abstract. We are interested in the structure of starlike bodies. Topological and smooth
classifications of such bodies in the infinite-dimensional spaces are given. This involves an
approximation of convex sets by smooth convex bodies. Some finite-dimensional examples
are also discussed.
This is a preliminary report; the details will appear elswhere.
A closed subset A of a Banach space X is a starlike body if its interior int A is nonempty
and there exists a point x0 ∈ int A such that every ray emanating from x0 meets ∂A, the
boundary of A, at most once. With the use of suitable translation, we can always assume
(and we will do so) that x0 = 0 is the origin of X.
For a starlike body A, we define the characteristic cone of A as
ccA = {x ∈ X|rx ∈ A for all r > 0},
and the Minkowski functional of A as
1
µA (x) = inf{λ > 0| x ∈ A}
λ
for all x ∈ X. It is easily seen that for every starlike body A its Minkowski functional µA is
a continuous function which satisfies µA (rx) = rµA (x) for every r ≥ 0 and µ−1
A (0) = ccA.
Moreover, A = {x ∈ X|µA (x) ≤ 1}, and ∂A = {x ∈ X | µA (x) = 1}. Conversely, if
ψ : X → [0, ∞) is continuous and satisfies ψ(λx) = λψ(x) for all λ ≥ 0, then Aψ = {x ∈
X | ψ(x) ≤ 1} is a starlike body. More generally, for a continuous function ψ : X → [0, ∞)
such that ψx (λ) = ψ(λx), λ > 0, is increasing and sup ψx (λ) > ε for every x ∈ X \ ψ −1 (0),
the set ψ −1 ([0, ε]) is a starlike body whose characteristic cone is ψ −1 (0). Starlike bodies
that are convex are called convex bodies. For a convex body U , ccU is always a convex set,
Typeset by AMS-TEX
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DANIEL AZAGRA AND TADEUSZ DOBROWOLSKI
but in general the characteristic cone of a starlike body is not convex. We will say that A
is a C p (or, real-analytic) smooth starlike body provided its Minkowski functional µA is C p
smooth (or, real-analytic) on the set X \ ccA = X \ µ−1
A (0). Finally, two (smooth) starlike
bodies A, B in a Banach space X are relatively homeomorphic (relatively diffeomorphic)
if there exists a self-homeomorphism (diffeomorphism) g : X → X so that g(A) = B.
Starlike bodies often appear in nonlinear functional analysis as natural substitutes of
convex bodies or in connection with polynomials, more precisely, for every n-homogeneous
polynomial P : X → R the set {x ∈ X|P (x) ≤ c}, c > 0, is either a (real-analytic) starlike
body or its complement is an interior of such a body (see [AD]). It is therefore reasonable
to ask to what extent the geometrical properties of convex bodies are shared with the more
general class of starlike bodies. In [AD] the question of whether James’ theorem on the
characterization of reflexivity (one of the deepest classical results of functional analysis)
is true for starlike bodies was answered in the negative. In [AC] it was shown that the
boundary of a smooth Lipschitz bounded starlike body in an infinite-dimensional Banach
space is smoothly Lipschitz contractible; furthermore, the boundary is a smooth Lipschitz
retract of the body. Here, we deal with the question as to what extent the known results
on the topological classification of convex bodies can be generalized for the class of starlike
bodies.
In [K], Klee gave a topological classification of the convex bodies of a Hilbert space. This
result was generalized for every Banach space with the help of Bessaga’s non-complete norm
technique (see [BP]). To get a better insight in the history of the topological classification
of convex bodies the reader should consult the papers by Stocker [S], Corson and Klee
[CK], Bessaga and Klee [BK1, BK2], and Dobrowolski [Do1]. These results have recently
been sharpened to obtain a full classification of the C p smooth convex bodies in every
Banach space [ADo]. In its most general form the result on the topological classification
of (smooth) convex bodies reads as follows (see [ADo]); here, p = 0, 1, 2, ..., ∞, and “C 0
diffeomorphic” means just “homeomorphic”.
Theorem 1. Let U be a C p convex body in a Banach space X.
(a) If ccU is a linear subspace of finite codimension (say X = ccU ⊕ Z, with Z finitedimensional), then U is C p relatively diffeomorphic to ccU + BZ , where BZ is an
STARLIKE BODIES
27
Euclidean ball in Z.
(b) If ccU is not a linear subspace or ccU is a linear subspace such that the quotient
space X/ccU is infinite-dimensional, then U is C p relatively diffeomorphic to a
closed half-space (that is, {x ∈ X | x∗ (x) ≥ 0}, for some x∗ ∈ X ∗ ).
Let us discuss to what extent this result can be generalized for (smooth) starlike bodies.
The following simple example shows that the assertion (b) of Theorem 1 is not true for
starlike bodies whose characteristic cones are not convex sets.
Example 1. Let A = {(x, y) ∈ R2 |x2 y 2 ≤ 1}. It is clear that A is a (real-analytic) starlike
body in R2 , whose characteristic cone is the union of the coordinate axes. Hence A, having
its boundary disconnected, cannot be relatively homeomorphic to a half-plane in R2 . However, every two (smooth) starlike bodies with the same characteristic cone are relatively homeomorphic (diffeomorphic). Though this fact is elementary, the proof of the
smooth case must be done with some care. The real-analytic counterpart of this fact is
unknown to us.
Proposition 1. Let X be a Banach space, and let A1 , A2 be C p smooth starlike bodies
such that ccA1 = ccA2 . Then there exists a C p diffeomorphism g : X → X such that
g(A1 ) = A2 , g(∂A1 ) = ∂A2 , and g(0) = 0. Moreover, g(x) = η(x)x, where η : X → [0, ∞),
and hence g preserves the rays emanating from the origin.
As said above, it is impossible to extend Theorem 1(b) to the class of starlike bodies.
The variety of the characteristic cones of (unbounded) starlike bodies is enormous. If one
wants to stick with the Bessaga-Klee classification scheme then the best result one can aim
at is the assertion of Theorem 1 for the class of starlike bodies whose characteristic cones
are convex sets.
We will state such a result, which relies on the following proposition, which might be
of independent interest in the theory of smoothness in Banach spaces, and which implies
that every closed convex cone in a separable Banach space can be regarded both as the
characteristic cone of some C ∞ smooth convex body and as the set of zeros of a C ∞ smooth
convex function. We say that a nonempty subset C of a Banach space X is a cone (resp.,
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DANIEL AZAGRA AND TADEUSZ DOBROWOLSKI
a cone over a set K) provided [0, ∞)C = C (resp., C = [0, ∞)K). The cone C is proper if
C = X.
Proposition 2. For every proper closed convex set C in a separable Banach space X there
exists a C ∞ smooth convex function f : X −→ [0, ∞) so that f −1 (0) = C and f (x) = 0
for all x ∈ X \ C. Moreover, when C is a cone, U = f −1 ([0, 1]) is a C ∞ smooth convex
body in X so that ccU = C.
Now we have arrived at the following generalization of Theorem 1.
Theorem 2. Let A be a C p starlike body in a separable Banach space
X. Assume that ccA is a convex subset of X.
(a) If ccA is a linear subspace of finite codimension (say X = ccU ⊕ Z, with Z finitedimensional), then A is C p relatively diffeomorphic to ccA + BZ , where BZ is a
Euclidean ball in Z.
(b) If ccA is either not a linear subspace or else ccA is a linear subspace such that the
quotient space X/ccA is infinite-dimensional, then U is C p relatively diffeomorphic
to a closed half-space.
In the case p = 0, the assertions (a) and (b) hold for all Banach spaces X.
Proof. To obtain (a) it is enough to apply Proposition 1 for A1 = A and A2 = ccA + BZ .
To obtain (b) write C = ccA for the closed convex cone of X. By Proposition 2, there
exists a C ∞ smooth convex body U so that ccU = C = ccA. Then, by Proposition 1,
the starlike bodies U and A are C p relatively diffeomorphic. On the other hand, by the
assumption, ccU = C is either not a linear subspace or else is a linear subspace such that
dim(X/C) = ∞. Now, by Theorem 1(b), U is C p relatively diffeomorphic to a closed
half-space, and hence so is A.
In the case p = 0, it is easy to see that, for every closed convex cone C ⊂ X, the set
U = C + B, where B is the unit ball of X, is a closed convex body so that C = ccU .
Hence, the above argument applies.
It is natural to ask whether, for starlike bodies A and B with homeomorphic boundaries
∂A and ∂B, A and B are relatively homeomorphic.
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29
The following theorem, answering this question in the affirmative, provides a full classification of starlike bodies in terms of the homotopy type of their boundaries in infinitedimensional Banach spaces.
Theorem 3. Let X be a Banach space and let A, B be starlike bodies in X with boundaries
∂A and ∂B. The following statements are equivalent:
(1) ∂A has the same homotopy type as ∂B;
(2) ∂A and ∂B are homeomorphic;
(3) A and B are relatively homeomorphic.
The proof involves infinite-dimensional topology, see [BP]. The bodies A and B, and
their boundaries ∂A and ∂B are so-called Hilbert manifolds. Since A and B are contractible, in fact, they are homeomorphic to X. Moreover, ∂A and ∂B are the so-called
Z-sets in A and B, respectively. The fact that ∂A and ∂B have the same homotopy type
implies they actually are homeomorphic. By the homeomorphism extension theorem for
Z-sets, any homeomorphism h : ∂A → ∂B extends to a homeomorphism H of A onto B.
Finally, it is easy to extend H to a self-homeomorphism of X.
Starlike bodies in a Banach space X are, in some sense, in one-to-one correspondence
with closed subset K (open subsets U ) of the unit sphere S of X. Let A be a starlike body
in X. Let r : X \ {0} → S be the radial retraction. Clearly, S(A) = r(ccA \ {0}) is a closed
subset of S such that ccA = [0, ∞)S(A), the cone over S(A), and r(∂A) = S \ S(A) is an
open subset of S. As it is easily seen below, a closed subset K of S gives rise to a starlike
body whose characteristic cone is the cone over K.
Proposition 3. Let K be a closed subset of S. There exists a starlike body A = AK such
that S(A) = K. If X is separable and C p smooth, then we may require that the body A is
C p smooth as well.
Proof. Take any continuous function λ : S → [0, 1] with λ−1 (0) = K. Define ψ(x) =
x
xλ( x
) for x = 0 and ψ(0) = 0. We see that ψ : X → [0, ∞) is a positively homogeneous
continuous function with ψ −1 (0) = [0, ∞)K. It is enough to set A = ψ −1 ([0, 1]).
In the smooth case, if X is C p smooth, there exists a bounded C p smooth starlike
body whose charcteristic cone is {0} [DGZ]. Let µ stand for the Minkowski functional of
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DANIEL AZAGRA AND TADEUSZ DOBROWOLSKI
this body. Using the fact that X admits C p smooth partitions of unity, one can find a
continuous function λ : X → [0, 1] which is C p smooth off λ−1 (0) = [0, ∞)K. Define
x
) for x = 0 and ψ(0) = 0. Clearly, ψ : X → [0, ∞) is a positively
ψ(x) = µ(x)λ( µ(x)
homogeneous continuous function which is C p smooth off ψ −1 (0) = [0, ∞)K. Set A =
ψ −1 ([0, 1]). Remark 1. The smooth assertion holds true if one replaces the separability assumption by
the existence of C p smooth partions of unity. In the proof of Proposition 3, instead of using the functional µ, we could have used a
weak hilbertian norm ω on the separable space X, that is, a continuous norm of the form
ω(x) = T (x) that is determined by an injective continuous linear operator T : X → 2 .
In such a case, ω is real-analytic off ω −1 (0). If K is a compact subset of S, then K0 =
([0, ∞)K) ∩ Bω , where Bω is the unit ω-sphere, is also compact. Hence, T (K0 ) is compact
in 2 and, by [Do2], there exists a continuous function λ : Bω → [0, 1] that is real-analytic
off λ−1 (0) = K0 .
x
) for x = 0 and ψ(0) = 0, the set A = ψ −1 ([0, 1]) is a
Remark 2. Letting ψ(x) = ω(x)λ( ω(x)
real-analytic starlike body with ccA = [0, ∞)K. As a consequence, in a separable Banach
space, for every starlike body A with the locally compact ccA there exists a real-analytic
starlike body A0 with ccA0 = ccA.
We do not know whether this last statement holds for an arbitrary starlike body A.
However, if ccA is weakly closed, then we can find a weak hilbertian norm ω so that ccA
is ω-closed. We can then construct a continuous function λ : Bω → [0, 1] that is C ∞ off
λ−1 (0) = ccA ∩Bω . Since the characteristic cone of a weakly closed starlike body is weakly
closed, we have the following:
Remark 3. For a starlike body A in a separable Banach space, which is closed in the weak
topology, there exists a C ∞ starlike body A0 with ccA = ccA0 .
For a closed set K ⊂ S, all (smooth) starlike bodies of the form AK are relatively
(diffeomorphic) homeomorphic. As a consequence of Theorem 3, we have:
Corollary 1. For two closed sets K1 , K2 ⊂ S in an infinite-dimensional Banach space X,
STARLIKE BODIES
31
the starlike bodies AK1 and AK2 are relatively homeomorphic if and only if the complements
S \ K1 and S \ K2 have the same homotopy type.
Proof. This is a consequence of Theorem 3 because the boundary of AKi is homeomorphic
to S \ Ki , i = 1, 2. It is unknown what necessary and sufficient conditions for Ki , i = 1, 2 one has to impose
in order for their complements in S to have the same homotopy type. (Since the sphere S
is homeomorphic to X, we can replace S by X.) If K is a Z-set in S (e.g., K is compact),
then the complement of K is homeomorphic to S; hence, in such a case, AK is relatively
homeomorphic to the unit ball. If K1 is a one-point set and K2 is a small closed ball
intersected with S, then K1 is a Z-set, while B2 is not a Z-set, but the complements of K1
and K2 have the same homotopy type (they are contractible), and therefore AK1 and AK2
are relatively homeomorphic (with the unit ball). The following simple example shows
that the contractibility of K1 and K2 does not suffice to obtain the same homotopy type
of their complements.
Example 2. Let K1 ⊂ S be a one point set and K2 = S ∩ X0 , where X0 is a codimension
1 vector subspace of X. Then, K1 and K2 are contractible, but the complement of K2
is disconnected, while the complement of K1 is contractible (even homeomorphic to X).
We see that AK1 is relatively homeomorphic to the unit ball in X, while ccAK2 = X0
and, consequently, AK2 is relatively homeomorphic to X0 × [−1, 1], which, in turn, (having
disconnected boundary in X0 × R) is not homeomorphic to the unit ball in X. Since, for a Zσ -set Z (that is, Z is a countable union of Z-sets) in S, the spaces S \ Z
and S are homeomorphic, one can hope that if K1 and K2 have the same homotopy type
modulo Zσ -set, then the complements of Ki , i = 1, 2, have the same homotopy type. (Two
closed sets P1 , P2 are meant to have the same homotopy type modulo Zσ -set if there are
closed sets Pi ⊂ Pi , i = 1, 2, such that Pi , i = 1, 2, have the same homotopy type and
both P1 \ P1 and P2 \ P2 are Zσ -sets.) This, however, is not the case because K1 and K2
of Example 2 have the same homotopy type modulo Zσ -set.
The finite-dimensional case. Below we provide several examples showing that Corollary 1 cannot be extended in any reasonable way for the finite-dimensional space X.
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DANIEL AZAGRA AND TADEUSZ DOBROWOLSKI
Example 3. Let S = S 1 and B be the unit sphere and the unit ball in X = R2 , respectively. Consider two compacta K1 and K2 in S; K1 is a copy of an infinite convergent
sequence space and K2 is a copy of the Cantor set. Then, the bodies AK1 and AK2 (having
their boundaries homeomorphic) are not homeomorphic.
To see this it suffices to notice that each AKi is homeomorphic to B \ Ki . It is then
clear that any non-isolated point of K1 has a basis of neighborhoods (in AK1 ) that can be
chosen to be topologically different from any neighborhood of any point of K2 . We can
obviously make those starlike bodies to be real-analytic, so an improvement in smoothness
is not any help.
In higher dimensions, one can provide more regular examples.
Example 4. Let S = S 2 be the unit sphere in X = R3 . Consider C1 = U1 ∪U2 ∪U3 , where
U1 = {(x, y, z) ∈ S||z| < 1/8}, U2 = {(x, y, z) ∈ S||z − 1| < 1/8}, and U3 = −U2 , and
C2 = U1 ∪ U2 ∪ U3 , where U3 = {x, y, z) ∈ S|z − 1/2| < 1/8, y > 0}. Letting Ki = S \ Ci ,
i = 1, 2, we see that the boundaries of the starlike bodies AKi (being homeomorphic to
Ci ) are homeomorphic. However, there is no homeomorphism of AK1 onto AK2 . In R4 , we have the following example.
Example 5. Let S = S 3 be the unit sphere in X = R4 . Let K be the (doubled) Fox-Artin
arc in S, that is, K is a topological arc whose complement is a contractible 3-manifold
which is not homeomorphic to R3 , see [Ru, p. 68]. Then, for a starlike body A = AK , ccA
is the cone over an arc, therefore, it is contractible. Moreover, AK is not homeomorphic
to a half-space in R4 though both bodies have contractible boundaries.
In general, for every n ≥ 4, the sphere S = S n−1 in X = Rn contains an open contractible (n − 1)-manifold U that is not homeomorphic to Rn−1 . One can take U to be
the so-called Whitehead manifold. In each dimension, there are continuum many pairwise
non-homeomorphic such objects. While the complement S 3 \ U is a continuum that is
not contractible, for n > 4, always one can pick U so that S n−1 \ U is a contractible
(n − 1)-manifold. To see this, let M be a contractible (n − 1)-manifold with non-simply
connected boundary; the existence of M is due to N.H.A. Newman for n > 5 (see [G]),
STARLIKE BODIES
33
and due to B. Mazur and V. Poenaru for n = 5. Gluing together two copies of M along
their boundaries we obtain the double space N , which is a topological copy of S n−1 (cf.
[AG, p. 2, items (4) and (9)]). The complement of one copy of M in N is just the interior
of the other copy, which yields a requested manifold U . Since U is not simply connected
at infinity, U is not homeomorphic to Rn−1 ; moreover, the manifold U , being the interior
of a contractible manifold, is itself contractible.
Example 5. Write K = S \ U . Any starlike body AK in Rn , n > 4, has both ccAK and
∂AK contractible. However, AK is not homeomorphic to a half-space in Rn . We wish to thank Craig Guilbault for his helpful observations that are included in
Examples 4-5.
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Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad
Complutense, Madrid, 28040, Spain
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DANIEL AZAGRA AND TADEUSZ DOBROWOLSKI
E-mail address: daniel [email protected]
Department of Mathematics, Pittsburg State University, Pittsburg, KS 66762
E-mail address: [email protected]